Unit 2 · Topic 07 · Number Play
Ms. Rao gave a simple rule: if a number is even, halve it; if it's odd, triple it and add 1. Repeat forever.
Kabir started with 6 and reached 1 in eight steps. Anaya started with 27 and kept going... and going... for over a hundred steps.
No matter which starting number anyone has ever tried, this strange bouncing sequence has always eventually landed on 1 — but nobody has ever proven it always must.
"Here's the rule," said Ms. Rao. "Pick any whole number. If it's EVEN, divide it by 2. If it's ODD, multiply it by 3 and add 1. Then do it again to whatever number you get, forever." Kabir tried 6: even, so 6/2=3. Then 3 is odd, so 3x3+1=10. Then 10 is even, so 10/2=5. Then 5 is odd, so 5x3+1=16. Then 16/2=8, 8/2=4, 4/2=2, 2/2=1. "I reached 1!" said Kabir. "After 8 steps."
Anaya tried a bigger number, 27. The sequence bounced up and down wildly — climbing as high as 9,232 at one point — before finally, after 111 steps, it also landed on 1. "Every single number anyone has tried," said Ms. Rao, "eventually reaches 1, no matter how wild the ride gets in the middle."
"So it's PROVEN that every number reaches 1?" asked Kabir. "That's the surprising part," said Ms. Rao. "It's been checked by computer for enormous numbers of starting values — trillions of them — and it has never once failed. But nobody has ever proven, using pure logic, that it MUST work for every possible number, forever. This is called the Collatz Conjecture — a 'conjecture' is a mathematical guess that looks true but hasn't been formally proven."
Anaya found this fascinating. "So a rule this simple — just halve or triple-and-add-1 — creates a mystery that's stumped mathematicians for decades?" Ms. Rao nodded. "That's exactly what makes it famous. Simple rules don't always have simple explanations, and sometimes the simplest-looking questions in mathematics turn out to be the hardest ones to answer for certain."
The Collatz rule: if a number is even, divide it by 2; if it is odd, multiply by 3 and add 1.
Applying this rule repeatedly to any starting whole number produces a 'Collatz sequence.'
Every starting number ever tested eventually reaches 1, even after wildly climbing up and down first.
A conjecture is a mathematical statement believed true from evidence, but not yet formally proven for every possible case — the Collatz Conjecture remains unproven despite decades of testing.
Work out the full Collatz sequence starting from 9, step by step, until you reach 1.
Explain in your own words the difference between a mathematical conjecture and a proven theorem.
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